Critical scaling in hidden state inference for linear Langevin dynamics

TL;DR

This paper analytically investigates the inference of hidden states in linear Langevin dynamics, revealing phase transitions and critical scaling behavior in the error and relaxation times based on network parameters.

Contribution

It provides a detailed phase diagram and critical scaling analysis for hidden state inference in stochastic linear networks with Gaussian couplings.

Findings

Identification of critical regions where error and relaxation time diverge

Derivation of the phase diagram in parameter space

Analysis of scaling behavior near critical points

Abstract

We consider the problem of inferring the dynamics of unknown (i.e. hidden) nodes from a set of observed trajectories and study analytically the average prediction error and the typical relaxation time of correlations between errors. We focus on a stochastic linear dynamics of continuous degrees of freedom interacting via random Gaussian couplings in the infinite network size limit. The expected error on the hidden time courses can be found as the equal-time hidden-to-hidden covariance of the probability distribution conditioned on observations. In the stationary regime, we analyze the phase diagram in the space of relevant parameters, namely the ratio between the numbers of observed and hidden nodes, the degree of symmetry of the interactions and the amplitudes of the hidden-to-hidden and hidden-to-observed couplings relative to the decay constant of the internal hidden dynamics. In particular, we identify critical regions in parameter space where the relaxation time and the inference error diverge, and determine the corresponding scaling behaviour.